tnt 



Abridged 



HG 1628 

1888 
Copy 1 



ABRIDGED 



I 

| INTEREST TABLES, 



| 

! 
I 



BY 



M. L.^EDMUNDS. 



5^. 



I 



I 

^ 169-171 Second Street. 



PORTLAND, OREGON: 

DAVID STEEL. SUCCESSOR TO HIMES THE PRINTER, 



1888. 



^^*^^3^^^ 



^"1. . ^<S)@: 



ABRIDGED 



'INTEREST TABLES, 



BY 



M. L. EDMUKDS. 



^>^v 



PORTLAND, OREGON : 

DAVID STEEL, SUCCESSOR TO HIMES THE PRINTER, 

169-171 Second Street. 
1888. 



<(S(Q>^ 







Entered according to Act of Congress, in the year iS86, by 

MILTON L. EDMUNDS, 
In the Office of. the Librarian of Congress, at Washingtoti. 



INTRODUCTION. 



The importance of a method that can be 
readily applied in the calculation of inter- 
est, has led to the exercise of considerable 
ingenuity in order to discover the shortest 
and simplest rule in practice. The object 
of this work is to present a method for 
computing interest, not only brief, but one 
that will give correct interest; this being a 
feature in which most methods are deficient 
in consequence of reckoning time incor- 
rectly. It may be readily seen that an error 
arises in the use of all methods for cal- 
culating interest, whereby the month is 
reckoned at 30 days, and consequently the 
year at 360 days; hence the objection to 
the favorite six per cent, method, also to 
various other methods, which, by reckon- 
ing 360 days to the year, give, for fractional 
parts of a year, an amount of interest ex- 
ceeding the exact interest by the same ratio 
that 365 days exceed 360 days. 



INTRODUCTION. 



Exact interest, obtained by reckoning 
365 days to the year, is growing in favor 
with bankers and other business men, is 
the method of interest used by the Uuited 
States Government and by foreign corres- 
pondents, is the method of interest becom- 
ing the most popular, and which ultimately 
is destined to be in universal use. 

In solving problems in simple interest, 
the primary object is to find the interest on 
a given principal for a given time and rate. 
That method which is the most natural and 
simple in principle, is to find the interest 
for one year by multiplying the principal 
by the rate, and then multiplying this in- 
terest by the time in years. The objection 
to this method, heretofore, has been in the 
difficulty of multiplying by the time, which, 
given in months and days, has been con- 
sidered incapable of being reduced to con- 
venient fractional parts of a year. The 
method of interest presented in this work, 
by having all fractional parts of a year ex- 
pressed decimally r , enables us to follow the 
natural process, while at the same time it 
gives the shortest method possible for cal- 
culating exact interest — a desideratum 



INTRODUCTION. 5 

hitherto imsupplied by any treatise on the 
subject. 

There being 365 days in a year, it is im- 
possible to divide the year into months 
each containing an equal number of entire 
days, and therefore impracticable to reckon 
time in months. This difficulty may be 
obviated by using methods whereby in- 
terest is calculated for the number of days. 
Indeed, the only correct methods are those 
by which interest, for periods of time less 
than one year, is calculated for the exact 
number of days ; and the most practical 
method — a need which this work is de- 
signed to supply — is that which by the 
most natural process, with the least amount 
of labor, will give exact interest. 

The amount of table work, not aggre- 
gating one-half page, all of which should 
be thoroughly committed to memory, forms 
a desirable feature of this method ; namely, 
the ease and rapidity by which we are en- 
abled to compute time and to reduce days 
to decimal years. It may also be observed 
that if such periods of time as are in fre- 
quent use have their decimal years mem- 
orized, the computation of interest for these 



INTRODUCTION. 



periods becomes susceptible of easy and 
rapid calculation. 

The method is conveniently treated un- 
der three cases, viz.: To find the time; to 
express the time decimally ; and to find the 
exact interest. To this is appended a gen- 
eral rule, also a variety of problems illus- 
trating the process of obtaining, and mul- 
tiplying by, the decimal years. 

Having prosecuted the work with the 
view ot facilitating the calculation of in- 
terest, the author now submits his method 
to the consideration of those whose avoca- 
tions demand a practical treatise on this 
important subject, and leaves whatever 
merit the method deserves to the decision 
of those competent to judge. 

M. L. Edmunds. 



ABRIDGED INTEREST TABLES. 



I. TO FIND THE TIME. 

In order to compute time readily, the 

following table, which gives the number of 

days in the year previous to the first day of 

each month, should be committed to 

memory : 

January 0. July 181. 

February 31. August 212. 

March..' 59. September 243. 

April 90. October 273. 

May 120. November 304. 

June 151. December 334. 

To find the difference of time between 

two dates, we first find the day of the year 

of each date by adding the day of the mo?ith 

of each date respectively to the numbers in 

the table corresponding to the given months, 

and then subtract the day of the year of 

the former date from the day of the year 

of the latter date, to find the difference of 

time in davs between two dates in the same 



8 ABRIDGED INTEREST TABLES. 

year; or, subtract the day of the year of 
the former date from 365 and add the re- 
mainder to the day of the year of the latter 
date, if the dates are in consecutive years, 
and the time less than one year; or, deter- 
mine the number of entire years, and then 
reckon the exact number of days remain- 
ing, by the foregoing r ales, if the time ex- 
ceeds one year; and add 1 to the number 
of days found by the table, in passing over 
February in leap year. 

Example 1. — Should you want to find the 
day of the year corresponding to March 10th, 
determine the number of days in the year pre- 
vious to the first day of March, which is shown 
by the table to be 59, to which add the day of 
the month, and you find March 10th to be the 
69th day of the year. 

Example 2.— To find the difference of time 
between February 12th, the 43d day of the year, 
and July 20th, the 201st day of the year, take the 
difference between 201 and 43, which is 158, the 
difference of time in days. 

Example 3. — The difference of time in days 
between November 15th, the 319th day of the 
year, and February 10th, the 41st day of the 
year following, is found by adding the difference 
between 365 and 319, which is 46, the number of 
days from November 15th to the close of the 
year, to 41, which gives 87 days. 



ABRIDGED INTEREST TABLES. 9 

II. TO EXPRESS THE TIME DECIMALLY. 

Since each day is 1 -365th of a year, there 
will be as many 365ths of a year in any 
given time as there are days, and the frac- 
tional part of a year thus represented may 
be reduced to a decimal year by annexing 
ciphers to the number of days and divid- 
ing by 365; the quotient thus obtained 
will be the time expressed decimally. 

Example. — Reduce 97 days to a decimal 
year. 

Solution.— 97 days equal 97-3G5ths of a year ; 
and 97.6 divided by 365 equals .265+ ; hence 97 
days equal .265+ of a year. 

To facilitate the process of reducing days 
to decimal years, commit to memory the 
following table: 

365x1= 365 365x4 = 1460 365x7 = 2555 
365x2= 730 365x5=1825 365x8 = 2920 
365x3=1095 365x6=2190 365x9=3285 

THE DECIMAL YEAR. 

In reducing days to decimal years, we annex ciphers 
to the number of days and divide by 365. The division 
will in most cases result in decimals which do not ter- 



10 ABRIDGED INTEREST TABLES. 

III. TO FIND THE EXACT INTEREST. 

Multiplying the principal by the rate of 
interest gives the interest for one year, and 
this interest multiplied by the time in years 
gives the required interest. 

The process of multiplying by the time, 
when expressed decimally, is performed by 
multiplying the interest for one year by 

minate, but, when expanded sufficiently far, will result 
in a series of figures called the repetend, which will con- 
stantly repeat in the same order. Such decimals are 
called circulating decimals, and those repetends in 
which the terms of the first half are respectively equal 
to 9 minus the corresponding terms of the second half, 
are called complementary repetends. The repetend 
of any circulating decimal year is complementary and 
consists of eight terms, and may be indicated by placing 
a period over the first and the last figures. 

.Let the reduction of 97.0-^-385 be continued five deci- 
mal places, and we have 2, the finite or non-repeating 
part ot the decimal, and 6575, the first half of the repetend. 
Subtracting the terms of the first half of the repetend re- 
spectively from 9 gives 3424, the terms of the last half, and 
we have the mixed circulate .265753424. whose repetend is 
complementary. It is therefore evident that in making 
such redaction, or in memorizing a decimal year, it is 
unnecessary to continue the reduction or the memor- 
izing further than is required to determine the first half 
of the repetend, since any number of terms following 
may be determined from the first half of the repeating 
part. 

When the number of days is 73, or a multiple of 73, 
the corresponding decimal year terminates with tenths. 
When the number of days is 5, or a multiple of 5, the 
corresponding decimal year results in a circulate whose 
repetend begins with the first term of the decimal. All 
other decimal years are circulates whose repetends be- 
gin with the second term of the decimal. 

The following table, consisting of nine series of figures, 
contains the repetend of the decimal found by reducing 



ABRIDGED INTEREST TABLES. 11 

the number of entire years, and each dec- 
imal division of the interest for one year 
by the corresponding decimal part of the 
given time, and taking the sum of these 
products. 

This process enables us to contract each 
product to the required denomination, and 
to reject all products of a lower denomina- 
tion than required in the entire product. 



any number of days to a decimal year, observing, how- 
ever, that repetends may begin with any figure of any 
series given in the table, and when beginning with any 
figure other than the first it will end with that part of the 
series preceding the figure with which it begins. Since 
there is but one 2 followed by a 4, but one 6 followed by a 
3, but one 7 followed by a 9, etc., in all of the series, it will 
be necessary to determine only two figures of the repe- 
tend of any circulating decimal year in order to know the 
whole repetend ; provided, that the table is committed 
to memory. The first two figures of a repetend may 
usually be calculated mentally, and from these, by 
means of the table, the whole repetend is at once appar- 
ent. To study the table, read each series round and 
round, as though the figures were arranged in a circle, 
until thoroughly memorized, having no particular place 
of beginning or of ending, as each figure will at various 
times be the first figure of a repetend. Remembering 
that any four successive figures of a series is the comple- 
ment of the remaining four renders the memorizing 
much easier : 

10958904 15068,193 20547945 
12328767 16438356 24657.534 
13698630 17808219 27397260 

As the number of decimal places ordinarily required 
is from three to five, the above principles of circulates 
are employed to expedite the process ot reduction only 
when interest is required on extremely large amounts, 
or when decimals are to be memorized. 



12 ABRIDGED INTEREST TABLES. 

Example. — Required the interest of 
$3987, for 2 years and 316 days, at 5 per 
cent. 

Remark.— 2 years and 316 days equal 2.86575 + years. 
OPERATION. 

$3987=Principal. 
.05=Rate. 



199. 35=1 nterest for one year. 
57568. 2=Time expressed decimally. 

398.70=Interest for 2 years. 
159.48=Interest for 8 tenths of a year. 
11.96=Interest for 6 hundredths of a year. 
1.00=Interest for 5 thousandths of a year. 
.14=Iaterest for 7 ten-thousandths of a year. 
l=Interest for 5 hundred-thousandths of a 
year. 



$571.29=Required interest. 

Solution.— Multiplying the principal, $3987, 
by the rate, .05, gives $199.35, interest for one 
year; and this interest divided by 10, 100, 1000, 
etc., which may be effected by moving the dec- 
imal point one, two, three, etc., places to the 
left, will give $19.93 + , $1.99, + $0.19 + , etc., which 
equal the interest for one-tenth of a year, one- 
hundredth of a year, one-thousandth of a 
year, etc. By writing the number of entire 
years— 2, and the terms of the decimal years, 
which are tenths of a year — 8, hundredths of a 



ABRIDGED INTEREST TABLES. 13 

year — 6, thousandths of a year — 5, etc., respect- 
ively under the right hand terms of the interest 
for one year — $199.35, one-tenth of a year — 
$19.93 + , one-hundredth of a year— $1.99 + , one- 
thousandth of a year — $0.19 + , etc., we have the 
terms of the decimal years written in an in- 
verted order, at the left of years, each properly 
written under that division of the year's interest 
to be multiplied by it. We multiply the interest 
for one year, one-tenth of a year, one-hundredth 
of a year, one-thousandth of a year, etc., respect- 
ively by the number of entire years, tenths of a 
year, hundredths of a year, thousandths of a 
year, etc., increasing each of these products by 
as many units as would have been carried to it 
from the product of the rejected terms, and one 
more when the second term towards the right 
in the product of the rejected terms is 5 or more 
than 5 ; and place the right hand terms of these 
products in the same column. The sum of these 
products gives the required interest. 

Note i. — The rejected terms are the denominations lower 
than cents in the interest for one year, tor one-tenth of a year, 
for one-hundredth of a year, etc. 

Note 2. — The terms of the decimal years must be extended 
one place farther to the lett than the terms of the number ex- 
pressing the interest for one year, in order to obtain the last 
product, which is equal only to the number of units that would 
have been carried from the product of the rejected terms. 



14 ABRIDGED INTEREST TABLES. 

GENERAL RULE. 

1. Multiply the principal by the rate of inter- 
est, to find the interest for one year, 

2. Write the number of entire years, when not 
exceeding 9, using a cipher if the time is less than 
one year, under that part of the interest for one 
year , generally cents, which is of the lowest de- 
nomination in the required interest. Reduce the 
number of days to decimal years and write the 
tenths ofayear, hundredths ofayear, thousandths 
of a year, etc., in a reverse order at the left of 
years, extending the terms of the decimal years, 
ivhen interminate, one place farther to the left 
than the terms of the number, expressing the in- 
terest for one year. If the number of entire years 
exceeds 9, write the excess under the 9 and use it 
in the same manner that the other multipliers are 
used. 

3. Regard the interest for one year divided by 
10, 100, 1000, etc., which will give the interest for 
one-tenth ofayear, one-hundredth ofayear, one- 
thousandth of a year, etc ; multiply these inter' 
ests respectively by the number of entire years, 
tenths ofayear, hundredths of a year, thousandth 8 



ABRIDGED INTEREST TABLES. 15 

of a year, etc., increasing each product by as 
many units as would have been carried to it from 
the product of the rejected terms, and one more 
when the second figure towards the right in the 
product of the rejected terms is 5 or more than 5 / 
and take the sum of these products for the re- 
quired interest. 

Note i. — In reducing days to decimal years, when solving 
problems, place the number ot days at the right of years, with 
the divisor, 365, at the right of days. Determine whether the 
first quotient figure is tenths of a year or hundredths of a year, 
by observing- whether one or t\* o ciphers must be annexed to 
the number of days in order to be divisible by 365. If more ci- 
phers are required they need not necessarily be annexed to the 
number of days, as the work may be made more concise by an- 
nexing them to the remainders only. The operation may be still 
further abbreviated by not writing the divisor 365, nor the pro- 
ducts ot 365 by the quotient figures, since the work can be car- 
ried on mentally, as in short division. 

Note 2. — Common interest may be calculated by the same 
process as exact interest; but in reducing days to decimal years, 
divide by 360 instead of 365. In this reduction the decimal, 
when interminate, results in a circulate whose repetend, con- 
sisting of but one figure, is readily found by observing when a 
quotient figure will constantly repeat. 

Note 3. — The following variation of the general rule will 
frequently be found more convenient: Multiply the principal by 
the rate, and this product by the exact number of days, and divide 
the result by 360 or 365, accordingly as common o* exact interest 
is required. 



16 



ABRIDGED INTEREST TABLES. 



ILLUSTRATIVE EXAMPLES. 



1. Required the in- 
terest of $225, for 2 years 
and 40 days, at 8 per 
cent. 



2. Required the in- 
terest of $256.75, for 93 
days at 5 per cent. 



OPERATION. 


OPERATION 


$225 


$256.75 


.08 


.05 



18.00 
5901.2 



40.0(365 



12.8375 
7452.0 



93.0(365 



36.00 

1.80 
.16 

1 


Lns. 

squired the in- 
$400, for 12 years 
days, at 12 per 

DERATION. 

22.00(365 
Ans. 


2.57 

.64 

5 

1 


$37.97 A 

3. R< 

terest of 
and 22 
cent. 

Ol 

$400 
.12 


$3.27 Ans. 

4. Required the 

common interest of 

$460.25, for 5 years and 

300 days, at 7 per cent. 

OPERATION. 

$460.25 
.07 


48 00 

2060.9 

3 


32.2175 
3338.5 300.0(360 

161.09 

25.77 

.97 

.10 

1 


432 00 
144.00 

288 
1 


$578.89 


$187.94 



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